Quantum Clocks and Path Integrals: An Intuitive Approach to Quantum Tunneling Time

The quantum tunneling time has been long studied both theoretically and experimentally. However, a consensus on the best way to conceptualize and to calculate the quantity has not been reached. For example, naïve methods using expectation values of position and momentum operators give unphysical negative values of tunneling time for certain wave packets. Other existing methods have various problems, such as affecting the dynamics of the system. Motivated by Page, Wootters, and Deutsch's approach to observable time, we employ the use of quantum clocks and a path integral method to study the problem of quantum transition and tunneling times. We envisage a qubit as it transitions between two quantum states coupled to a quantum clock with finite resolution. By assigning each classical path a probability and characteristic transition time by the quantum clock, we then perform a path sum to determine a characteristic transition time for the qubit. This intuitive method can also be generalized from transition time to tunneling time as the density of states of the observable becomes continuous.